For the countable case, this is very easy: let $\alpha$ be countable (so $cf(\alpha)=\omega$) and additively decomposable. Then $\alpha$ is a limit, and we can decompose $\alpha$ into its "odd" and "even" bits: that is, there is a function $f: \alpha\rightarrow\alpha$ which is injective and order-preserving, such that $f(0)>0$, and $\alpha - im(f)\cong\alpha$. By iterating $f$, we get the desired partition of $\alpha$: $Y_0=\alpha- im(f)$, $Y_{n+1}=\alpha - (f^{n+1}(\alpha)\cup \bigcup_{i\le n} Y_i).$ (The condition $f(0)>0$ is used to guarantee that the minima of the $Y_i$ are cofinal in $\alpha$.)
For uncountable $\alpha$, I find it easier to switch to a slightly less concrete picture. Let $Y_i=\alpha\times\{i\}$ for $0\le i<cf(\alpha)$, and let $(\gamma_i)_{i<cf(\alpha)}$ be an increasing cofinal subset of $\alpha$ with $\gamma_0=0$. Let $h: cf(\alpha)^2\cong cf(\alpha)$ be a pairing function on the cofinality of $\alpha$ such that $$(*): \,\, \text{ $\{h(0, i): i<cf(\alpha)\}$ is cofinal in $\alpha$,}$$ and for $i=h^{-1}(j, k)$ let $$L_i=[\gamma_j, \gamma_{j+1}]\times\{k\}.$$ for $i<cf(\alpha)$. Then by induction, $\sum_{i<j} L_i<\alpha$ for each $j<\alpha$: at limit stages we use the cofinality of $\alpha$, and at successor stages we use the additive indecomposability of $\alpha$. On the other hand, we clearly have $\alpha\le\sum_{i<cf(\alpha)}L_i$, and so $$ \alpha\cong\sum_{i<cf(\alpha)}L_i.$$ This representation on $\alpha$ then pulls back to a collection of maps from the $T_i$, which - by our condition $(*)$ on $h$ - clearly satisfies that the images of the minima are cofinal.
